| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2020 |
| Session | March |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Symmetry property of PDF |
| Difficulty | Standard +0.3 This is a straightforward continuous probability distribution question requiring standard integration techniques and exploiting symmetry of an even function. Part (a) is routine integration, part (b) involves setting up and simplifying an integral equation (algebraically intensive but mechanical), and part (c) uses symmetry of the PDF about x=0. While multi-part, each step follows standard S2 procedures without requiring novel insight. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf |
5 Bottles of Lanta contain approximately 300 ml of juice. The volume of juice, in millilitres, in a bottle is $300 + X$, where $X$ is a random variable with probability density function given by
$$f ( x ) = \begin{cases} \frac { 3 } { 4000 } \left( 100 - x ^ { 2 } \right) & - 10 \leqslant x \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Find the probability that a randomly chosen bottle of Lanta contains more than 305 ml of juice.
\item Given that $25 \%$ of bottles of Lanta contain more than $( 300 + p ) \mathrm { ml }$ of juice, show that
$$p ^ { 3 } - 300 p + 1000 = 0$$
\item Given that $p = 3.47$, and that $50 \%$ of bottles of Lanta contain between ( $300 - q$ ) and ( $300 + q$ ) ml of juice, find $q$. Justify your answer.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2020 Q5 [9]}}